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with respect to a? is equal to the function itself - that is, & is the
only function which is equal to its derivative.*
The concept of transcendence* is extremely simple, also
extremely important. Any root of an algebraic equation whose
coefficients are rational integers (0, ~1 i2, . - -) is called an
algebraic number. Thus V  1, 2-TS are algebraic numbers,
because they are roots of the respective algebraic equations
#3 j^ i  o, 5Qx  139 = 0, in which the coefficients (1,1 for
the first; 50, - 139 for the second) are rational integers. A
'number' which is not algebraic is called transcendental. Other-
wise expressed, a transcendental number is one which satisfies
no algebraic equation with rational integer coefficients.
Xow, given any 'number' constructed according to some
definite law, it is a meaningful question to ask whether it is
algebraic or transcendental. Consider, for example, the follow-
ing simply defined number,
_i +JL J.1 -i-    *    -L-    1    4-
10      102   '   10*  '   10-*  '   10120      "" * *
in which the exponents 2, 6, 24, 120, ... are the successive
'factorials', namely 2 = 1x2, 6 = 1x2x3, 24 =1x2x3
X 4,120 = 1x2x3x4x5,..., and the indicated series
continues 6to infinity* according to the same law as that for the
terms given. The next term is -  ; the sum of the first three
terms is ! + -01 -f -000001, or -110001, and it can be proved
that the series does actually define some definite number which
is less than -12. Is this number a root of any algebraic equation
with rational integer coefficients? The answer is no, although to
prove this without having been shown how to go about it is a
severe test of high mathematical ability. On the other hand, the
number defined by the infinite series
--h-4- +  4-
10*      103      1011      1014
* Strictly, ae*t where a does not depend upon #, is the most general,
but the 'multiplicative constant* a is trivial here.